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The problem provided asks us to use synthetic division and the remainder theorem to find the remainder when \( f(x) = x^3 - 4x^2 + x - 5 \) is divided by \( x - 1 \).

### Step 1: Set up for Synthetic Division
- The divisor is \( x - 1 \), which means we will divide by \( 1 \) (the root of \( x - 1 = 0 \)).
- The coefficients of \( f(x) \) are \( [1, -4, 1, -5] \).

### Step 2: Synthetic Division Process
We will divide \( 1 \) into the polynomial:

\[
\begin{array}{r|rrrr}
1 & 1 & -4 & 1 & -5 \\
  &   & 1  & -3 & -2 \\
\hline
  & 1 & -3 & -2 & -7 \\
\end{array}
\]

### Explanation of Each Step:
- Bring down the first coefficient (1).
- Multiply 1 by \( 1 \) and add it to \( -4 \), giving \( -3 \).
- Multiply \( -3 \) by \( 1 \) and add it to \( 1 \), giving \( -2 \).
- Multiply \( -2 \) by \( 1 \) and add it to \( -5 \), giving \( -7 \).

### Step 3: The Remainder Theorem
The remainder theorem states that the remainder when dividing \( f(x) \) by \( x - c \) is simply \( f(c) \). In this case, \( c = 1 \), so the remainder is \( f(1) \).

Plug \( 1 \) into the polynomial:

\[
f(1) = 1^3 - 4(1^2) + 1 - 5 = 1 - 4 + 1 - 5 = -7
\]

### Conclusion
The remainder when \( f(x) \) is divided by \( x - 1 \) is \( \boxed{-7} \).

Would you like any further explanation or have any questions?

Here are 5 related questions to deepen your understanding:
1. How does synthetic division work when dividing by \( x - c \) for any \( c \)?
2. What is the relationship between synthetic division and the remainder theorem?
3. How would you use synthetic division if the divisor was \( x + 2 \) instead of \( x - 1 \)?
4. What happens if the polynomial has missing terms (e.g., no \( x^2 \) term)?
5. Can synthetic division be used for polynomials of higher degrees (e.g., \( x^4 \))?

**Tip:** Always verify your remainder by plugging the divisor into the original polynomial using the remainder theorem!

Use synthetic division and the remainder theorem to find the remainder when f(x) is divided by x - c. f(x) = x^3 - 4x^2 + x - 5; x - 1.

Explore how to use synthetic division and the remainder theorem to find the remainder of f(x) = x^3 - 4x^2 + x - 5 when divided by x - 1. This problem involves polynomial division, synthetic division steps, and application of the remainder theorem. Suitable for students in Grades 9-12.

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